A registered result in its source context

QGEO.REAL.01

Original ledger classification: Exact [E] existence/uniqueness/char identities + [X] quantified joint negative + [C] the TFPT readings (exact sympy; numeric prefilter with symbolic confirmation for the scan)

Original claim text

The real structure on the mu3-cover -- the v598 suspect constructed (one module, 18 checks, ~3 s; discovery probe real_structure_probe.py 18/18, verdict REAL-STRUCTURE-LANDED). EXISTENCE AND UNIQUENESS [E]: complex conjugation on the base fixes the punctures {1,-1} and swaps {i,-i} (an orientation-reversing D4 reflection); it must act semi-linearly, R(v) = M conj(v), sending each cusp twist to the INVERSE of its reflected partner, R T_k R^-1 = T_(5-k)^-1. The intertwining system has a 1-DIMENSIONAL solution space: R exists and is unique up to scale; the canonical normalization satisfies R^2 = +1 exactly (a genuine real structure, not quaternionic); all four twist intertwinings hold exactly. THE GRADING FUNCTOR [E]: Gamma(A) = M conj(A) M^-1 (the curve analogue of the compiler grading A -> Sigma A Sigma) closes the three v598 dictionary gaps at char level: V ~ G2 + G3 - Gamma(G3) with char x(x-1)(x-2); H ~ 1 + G2 - Gamma(G3) with char (x-1)^2(x-2) -- THE ANCHOR SPECTRUM (1,1,2); Q_+ ~ 1 + G2 + G3 - Gamma(G3) with char (x-1)(x-2)(x-3). THE CONSTANTS [E]: the Gamma-even/odd cusp elements E_k = G_k + Gamma(G_k), O_k = G_k - Gamma(G_k) carry the compiler constants: E2 spectrum {0,1,5} (the carrier g_car = 5), E4 {0,3,9} = {0, N_fam, N_fam^2}, O3 nilpotent (char x^3), O2 char x(x^2-5), O4 char x(x^2-27). THE JOINT NEGATIVE [X, quantified]: char-matching is NOT yet an algebra embedding -- in the +-1 Gamma-extended integer lattice (8 generators) there are exactly 4 U-candidates and 6 V-candidates, and ZERO pairs satisfy the joint compiler relations tr(UV) = 3 and char(U+V) = (x-1)(x^2-5x+3); the natural pair (G2, G2+G3-Gamma(G3)) gives char x(x^2-6x+4), an exact documented mismatch. The flag-adapted simultaneous-conjugation search remains the named next step; the Strategy-I kill criterion is NOT triggered. THE LORENTZ FORM [E, Strategy II]: the invariant hermitian form J (T_k^dagger J T_k = J for all four twists) is unique up to real scale, integral over Z[omega], with det J = 8 -- A MEMBER OF THE SHEET-DIAMOND DETERMINANT LIST (3,4,8,14,20,32) -- and char (x+2)(x^2-4x-4): LORENTZ SIGNATURE (1,2), matching the Paper-II rank-3 polarization signature AND the Chevalley-Weil Hodge prediction (h^{1,0}, h^{0,1}) of Strategy II simultaneously. GATE.QGEO does not move; no marker changes. [2026-08-01 XVI: the degree-2 Gamma-even sector closes the joint gap -- v600: the compiler pair (U, V) is realized (GL3-conjugation solved exactly, saturation index 81 on the R-fixed lattice); the v599 joint negative is thereby located precisely at degree 1; see QGEO.EMBED.01.]

Original classification detail

REAL-STRUCTURE-LANDED: R unique (R^2 = +1), four intertwinings exact; Gamma closes V/H/Q_+ at char level (H carries the anchor spectrum); constants 5/9/27 in the Gamma-decomposition; joint embedding open (4x6 candidates, 0 joint pairs); det J = 8, Lorentz (1,2); 18/18 checks

Dependencies as recorded

QGEO.DICT.01; QGEO.COVER.01; QGEO.INVOL.01; DIAMOND.PARA.SELFCODE.01; v566; v597; v598